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Implicit differentiation

Problem 2.805 · medium

The curve \( \displaystyle 4 x^{2} + 3 x y + 2 y^{2} = 3 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.

\[ 3 \]
The point is on the curve.Reviewed
Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
Reviewed
\[ \frac{d}{d x} \left(4 x^{2} + 3 x Y{\left(x \right)} + 2 Y^{2}{\left(x \right)}\right) = 8 x + \left(3 x + 4 Y{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + 3 Y{\left(x \right)} \]
Every y term picks up a factor dy/dx.✓ Proved
\[ \frac{- 8 x - 3 y}{3 x + 4 y} \]
Solve for dy/dx: minus F_x over F_y.Reviewed
\[ 5 \]
At the point.Reviewed

Answer \( \frac{dy}{dx} = \frac{- 8 x - 3 y}{3 x + 4 y}, \quad \left.\frac{dy}{dx}\right|_{(1,-1)} = 5 \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	sympy.idiff differentiates the curve on its own and agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The logic is sound and the steps are clear.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The logic is sound and the steps are clear.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies implicit differentiation, solves for dy/dx, and evaluates it at the given point. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-09-26

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/implicit_differentiation, checked 2026-10-13 with SymPy 1.14.0.